Finance Products and Markets

Lecture 3

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• Structured finance: problems and products

• Exotic options

• Univariate structured products

• Long/short volatility

• Multivariate structured products

• Long/short correlation

Structured finance: the questions

• A structured finance product is a security that includes one or more derivative contracts.

• The first question is whether the derivative contract is in the repayment plan or in the coupon plan

• The second question is who is writing the derivative, and who is buying it

• The third question is the sign of exposure to: i) underlying; ii) volatility; iii) correlation

A general rule of thumb

• If in the prospect of the security there is a clause stating that at some future time the value of a cash flow will be max(y, K), with K a given value, then that cash flow has an option in favour of the investor (the receiver). In this case the option increases the value of the security

• If the there is a clause stating that a cash flow will be min(y, K), then that cash flow has an option in favor of the issuer (the payer). In this case the value of the option decreases the value of the security.

Decomposition via put-call parity

• Option in favour of the receiver (investor) – Max( y, K) = y + max(K – y, 0)– Max ( y, K) =K + max(y – K, 0)– Long underlying, long volatility

• Option in favour of the payer (issuer)– Min( y, K) = y – max(y – K, 0)– Min ( y, K) =K – max(K – y, 0)– Long underlying, short volatility

Putable bonds

• Traditional example is the CTO (certificati del Tesoro con opzione): Italian government securities issued in the 80s with 6 year maturity retractable at 3 years

• Three years after issuance (time ) the value of the bond was

• CTO(,T;c) = max(P(,T;c), 1) = P(,T;c) + max(1 – P(,T;c),0) = 1 + max(P(,T;c) – 1,0)

• At time t, the value can be seen as a bond expiring at time T and a put option with exercise at (retractable) or a bond expiring a time and a call on a three year bond (extendible).

Callable bonds

• Callable bonds are so traditional and well known that they are not considered structured finance products

• At the call date (time ) the value of the bond is• CALLABLE(,T;c) = min(P(,T;c), 1)

= P(,T;c) – max(P(,T;c) – 1,0) = 1 – max(1 – P(,T;c),0)

• At time t, the value can be seen as a bond expiring at time T minus a call option with exercise at (retractable) or a bond expiring a time and the sale of o put option on a bond expiring at T (extendible).

Bermudan clauses

• In many callable bonds the callability clause is bermudan, meaning that it can be exercised on a set of given dates.

• Bermudan options are the intermediate case between European options (exercize at maturity) and American option (exercise by the date of maturity).

• Typical example: corporate bond with 15 year maturity callable every six months a10 years after issuance.

Convertibles

• In convertible bonds the repayment of the principal can be done in terms of cash or stocks (equity).

• In convertibles the choice of repayment is with the investor.

• In reverse convertibles the choice of repayment is with the issuer.

Convertible

• The repayment plan can be decomposed asmax(100, nS(T)) =

100 + n max(S(T) – 100/n, 0)

• The product includes n call options on the underlying with strike 100/n.

• In many cases, the options involved are endowed with the Bermudan clause.

Example

• The underlying: ENEL• Number of stocks: n = 12,5• Strike n = 100/12,5 = 8• Repayment

max(100, 12,5 Enel(T)) =

100 + 12,5 max(Enel(T) – 8, 0)• The value of the bond includes 12,5 call call

options with strike at 8 euro

Reverse convertible

• The repayment plan• min(100, nS(T)) =

100 – n max(100/n –S(T), 0)

• Il product includes a short position in n put options with strike 100/n.

• In many cases includes a very high coupon to attract investors and a barrier.

Example

• The underlying: ENEL• Number of stocks: n = 12,5• Strike n = 100/12,5 = 8• Repayment

min(100, 12,5 Enel(T)) =

100 – 12,5 max(8 – Enel(T), 0)• The value of the bond includes a short position

in 12,5 options with strike at 8 euros

Parallel interview

• Convertibile• Call option• To the investitor• Long call• Long underlying• Long gamma• Long vol

• Reverse convertible• Put option• To the issuer • Short put• Long underlying • Short gamma• Short vol

CoCos (Contingent Convertible)

• CoCos (contingent convertible): convertible bonds for which the investor can choose the payment only if the price has grown above a barrier before the maturity.

• CoCos for banks: convertible bonds issued by banks that can be converted into equity if the RWA (risk weighted assets) are falling below a given level. This kind of bonds is allowed as regulatory capital.

Reverse convertible

• Period: 1° Feb-1° Sept 2000• Coupon 22%, paid 01/09/2000• Repayment of principal in cash or in Telecom

stocks if two conditions apply – On 25/08/2000 Telecom stock quotes below 16.77

Euros– Btw 28/01/2000 and 25/08/2000 the price has

reached Telecom the threshold of 13.416 Euros

• Reverse convertible = ZCB – put (with barrier)

Barrier options

• Barrier options include a threshold, so that the final value

of the optiion depends on whether or not the underlying

asset (or some other underlying) has reached that value

(called barrier) during the lifetime of the option (or in

subperiod of it);

• Barrier options can be divided into:– Up barrier

– Down barrier

– Knock-in barrier

– Knock-out barrier

Barrier options: Knock-in

Down-and-in option

The option is activated

Up-and-in option

The option is acticvated

Barrier option: Knock-out

Down-and-out option

The option disappears

Up-and-out option

The option disappears

Parisian, parasian & co…

• In order to avoid market abuse, barrier options can be made more difficult to manipulate.

• Parisian options are activated if the price of the underlying remains below (or above) the barrier for more than a given period time without interruption or cumulated (cumulative parisian)

• Alternatively, the barrier can be compared with average values instead of point in time values (parasian).

Example – Parisian up

Equity linked note

• Assume a structure like this:– Investment with guarantee of the principal for 5

years – Coupon paid at maturity, linked to the

performance of a stock index • Questions:

– Who would buy it? And which options to include?

– What is the value of the product, both in the “host” bond and the derivative part

– What are the risk exposures? And how can it be handled?

Alternative investments

• The product allows to invest in the stock market over a long time horizon. The perspectives of profit are lined to: i) market and ii) volatility

• Alternatives: – Fund management with guarantee on the

principal (CPPI)– Options and warrants– Long term options (private banking) – Dynamic management with ETF or futures.

Structuring choices

• The degree of risk of the equity linked note can be altered in two ways. – Increase of the strike– Reduction of volatility

• The change of strike can be obtained in two ways: – Including a guaranteed return rg

– Including a participation rate • The payment at maturity will be: max(S(T),1 + rg)• The payoff can be decomposed as:

1 + rg + max[S(T) – (1 + rg )/ , 0]

Structuring choices

• Il order to reduce risk, volatility can be reduced by

– Using the average price as underlying asset (smoothing)

– Using the average price of different markets (diversification)

• This can be achieved by introducing exotic options: asian options and basket

Index-Linked Bond

• Consider the bond• Period: 31 July 2000 – 31 July 2004• Coupon and principal: paid at marturity• Fixed principal, coupon computed as the higher

between 6% and the average increase of end of quarter of a equally weighted portfolio of: Nikkei 225, Eurostoxx 50 e S&P 500.

• Index-Linked Bond = zero coupon + option

(asian call basket)

Asian options• Asian options use average of the price for the

underlying (average rate) or for the strike (average strike). In some cases averages are computed in discrete time with different frequencies.

• Valuation techniques:– Moment matching (Turnbull e Wakeman): the

distribution of the average is approximated with a log-normal distribution with same mean and variance

– Monte Carlo method: scenarios are generated for the sampling dates, the pay-offs for every path and the average is computed

Crash protection

• The investment horizon of this product can be perceived as too long. If the market decreases by a relevant amount, the option value gets to zero and the investor can remain locked-in in a low return investment.

• For this reason, the production could be enhanced by including the so-called crash protection clause. For this clause, if the value of the underlying decreases below a given percentage of the initial value, the new strike of the option is reset at that level.

Crash protection: valuation

• The value of the product is nowZCB + Call Ladder (S(t)/S(0), t; 1, h)

• We can isolate the value of the crash protection clause using – The replicating portfolio of the ladder option – The symmetry between in and out options

• We compute ZCB + Call(S(t)/S(0), t; 1, h) +

Down-and-In(S(t)/S(0), t;h, h) – Down-and-In(S(t)/S(0), t; 1, h)• The value of the crash protection clause is then given by

the difference between Down-and-In option with strike equal to the barrier and that with the original strike,

A different product

• We can think that investors would be more allured by a product that could generate income and cash flows through time more than by a product the paid the coupon in the end.

• We could think of a sequence of coupons like

Coupon (t + i) = max[S(t + i)/S(t + i – 1 ) – 1,0] • This way, the product would produce a cash flow

of interest equal to the appreciation of teh market, excluding losses.

Cliquet index-linked

• The new product can be represented as a sequence of coupons that are determined as a sequence of forward start options, that is a ratchet (cliquet)

• If one rules out the presence of dividends, the value of N coupons amounts to the sum of N at-the-money options with one year exercise.

• In the product with a single coupon at maturity the interest payment is a single at-the-money option with five year maturity.

Reverse cliquet (Vega bond)

• Assume a product that in N years pay a coupon defined as

Coupon = max[0, D + imin(S(t+i)/S(t+i–1) – 1,0)]

• In other terms, the coupon the coupon is made by an initial endowment D, from which negative changes of the market are subtracted in every period.

• This product is called vega bond for the exposure to volatility.

Multivariate digital notes(Altiplano)

• Assume that a coupon is determined (reset date) and paid at time tj.

• Assume a basket of n = 1,2 bonds, whose prices are Sn(tj).

• Denote Sn(t0) the reference prices (strike), typically recorded at the beginning of the contract.

• Denote Ij the indicator function taking value 1 if Sn(tj)/Sn(t0) > 1 for both assets and 0 otherwise.

• The coupon is a bivariate digital option, paying c

jc*ICedola

Bivariate digital note

• Investment horizon: March 2000 - March 2005 • Principal repaid at maturity• Coupon paid March 15 every year.

Coupon = 10% if (i = 1,2,3,4,5)Nikkei (15/3/200i) > Nikkei (15/3/2000) and Nasdaq 100 (15/3/200i) > Nasdaq 100 (15/3/2000)

Coupon = 0% otherwise

• Digital Note = ZCB + bivariate digital calls

Altiplano with memory

• Assume a coupon defined (reset date) and paid at time tj, and a sequence of dates {t0,t1,t2,…,tj – 1}.

• Assume a set of n = 1,2,…N assets, whose prices are Sn(ti).

• Denote B a barrier and Ii the monitor taking value 1 if Sn(ti)/ S(t0) > B for all bonds and 0 otherwise.

• The coupon of a Altiplano bond

wher c is a coupon and k is the guaranteed return.

11Cedola1

1

j-

iijj IcIkc*I

Everest

• Assume a coupon defined and paid at time T.• Assume a basket ofi n = 1,2,…N bonds, whose prices

are Sn(T).• Denote Sn(t0) the reference prices (strike), typically

recorded at inception of the contract all’origine del contratto, e usati come prezzi strike.

• The payoff is

max[min(Sn(T)/Sn(0),1+k] = = (1 + k) + max[min(Sn(T)/Sn(0) – (1+k),0]

with n = 1,2,…,N and minimun guaranteed return k.

Basket bond• Assume a coupon defined and paid at time T.• Assume a basket ofi n = 1,2,…N bonds, whose prices

are Sn(T).• Denote Sn(t0) the reference prices (strike), typically

recorded at inception of the contract all’origine del contratto, e usati come prezzi strike.

• The payoff is

max[Average(Sn(T)/Sn(0),1+k] = = (1 + k) + max[Average(Sn(T)/Sn(0) – (1+k),0]

with n = 1,2,…,N and minimun guaranteed return k.

Long/short correlation

• The sign of the exposure to correlation is linked to the presence in the product of clauses AND or OR for the pricing kernel of the derivative contracts embedded in the product.

• In Everest the sign is clear: it is a long position in correlation.

• Assume a product that pays a coupon given by the maximum of a set of appreciation rates if this is greater than a guaranteed return. The product is short correlation.